1. Field of the Invention
The present invention generally relates to a method of surveying earth formations in a borehole and, more specifically, to a method of and apparatus for independently determining the electrical resistivity and/or dielectric constant of earth formations during Measurement-While Drilling/Logging-While-Drilling and Wireline Logging operations.
2. Description of the Related Art
Typical petroleum drilling operations employ a number of techniques to gather information about earth formations during and in conjunction with drilling operations such as Wireline Logging, Measurement-While-Drilling (MWD) and Logging-While-Drilling (LWD) operations. Physical values such as the electrical conductivity and the dielectric constant of an earth formation can indicate either the presence or absence of oil-bearing structures near a drill hole, or xe2x80x9cborehole.xe2x80x9d A wealth of other information that is useful for oil well drilling and production is frequently derived from such measurements. Originally, a drill pipe and a drill bit were pulled from the borehole and then instruments were inserted into the hole in order to collect information about down hole conditions. This technique, or xe2x80x9cwireline logging,xe2x80x9d can be expensive in terms of both money and time. In addition, wireline data may be of poor quality and difficult to interpret due to deterioration of the region near the borehole after drilling. These factors lead to the development of Logging-While-Drilling (LWD). LWD operations involve collecting the same type of information as wireline logging without the need to pull the drilling apparatus from the borehole. Since the data are taken while drilling, the measurements are often more representative of virgin formation conditions because the near-borehole region often deteriorates over time after the well is drilled. For example, the drilling fluid often penetrates or invades the rock over time, making it more difficult to determine whether the fluids observed within the rock are naturally occurring or drilling induced. Data acquired while drilling are often used to aid the drilling process. For example, MWD/LWD data can help a driller navigate the well so that the borehole is ideally positioned within an oil bearing structure. The distinction between LWD and MWD is not always obvious, but MWD usually refers to measurements taken for the purpose of drilling the well (such as navigation) whereas LWD is principally for the purpose of estimating the fluid production from the earth formation. These terms will hereafter be used synonymously and referred to collectively as xe2x80x9cMWD/LWD.xe2x80x9d
In wireline logging, wireline induction measurements are commonly used to gather information used to calculate the electrical conductivity, or its inverse resistivity. See for example U.S. Pat. No. 5,157,605. A dielectric wireline tool is used to determine the dielectric constant and/or resistivity of an earth formation. This is typically done using measurements which are sensitive to the volume near the borehole wall. See for example U.S. Pat. No. 3,944,910. In MWD/LWD, a MWD/LWD resistivity tool is typically employed. Such devices are often called xe2x80x9cpropagation resistivityxe2x80x9d or xe2x80x9cwave resistivityxe2x80x9d tools, and they operate at frequencies high enough that the measurement is sensitive to the dielectric constant under conditions of either high resistivity or a large dielectric constant. See for example U.S. Pat. Nos. 4,899,112and 4,968,940. In MWD applications, resistivity measurements may be used for the purpose of evaluating the position of the borehole with respect to boundaries of the reservoir such as with respect to a nearby shale bed. The same resistivity tools used for LWD may also used for MWD; but, in LWD, other formation evaluation measurements including density and porosity are typically employed.
For purposes of this disclosure, the terms xe2x80x9cresistivityxe2x80x9d and xe2x80x9cconductivityxe2x80x9d will be used interchangeably with the understanding that they are inverses of each other and the measurement of either can be converted into the other by means of simple mathematical calculations. The terms xe2x80x9cdepth,xe2x80x9d xe2x80x9cpoint(s) along the borehole,xe2x80x9d and xe2x80x9cdistance along the borehole axisxe2x80x9d will also be used interchangeably. Since the borehole axis may be tilted with respect to the vertical, it is sometimes necessary to distinguish between the vertical depth and distance along the borehole axis. Should the vertical depth be referred to, it will be explicitly referred to as the xe2x80x9cvertical depth.xe2x80x9d
Typically, the electrical conductivity of an earth formation is not measured directly. It is instead inferred from other measurements either taken during (MWD/LWD) or after (Wireline Logging) the drilling operation. In typical embodiments of MWD/LWD resistivity devices, the direct measurements are the magnitude and the phase shift of a transmitted electrical signal traveling past a receiver array. See for example U.S. Pat. Nos. 4,899,112, 4,968,940, or 5,811,973. In commonly practiced embodiments, the transmitter emits electrical signals of frequencies typically between four hundred thousand and two million cycles per second (0.4-2.0 MHz). Two induction coils spaced along the axis of the drill collar having magnetic moments substantially parallel to the axis of the drill collar typically comprise the receiver array. The transmitter is typically an induction coil spaced along the axis of a drill collar from the receiver with its magnetic moment substantially parallel to the axis of the drill collar. A frequently used mode of operation is to energize the transmitter for a long enough time to result in the signal being essentially a continuous wave (only a fraction of a second is needed at typical frequencies of operation). The magnitude and phase of the signal at one receiving coil is recorded relative to its value at the other receiving coil. The magnitude is often referred to as the attenuation, and the phase is often called the phase shift. Thus, the magnitude, or attenuation, and the phase shift, or phase, are typically derived from the ratio of the voltage at one receiver antenna relative to the voltage at another receiver antenna.
Commercially deployed MWD/LWD resistivity measurement systems use multiple transmitters; consequently, attenuation and phase-based resistivity values can be derived independently using each transmitter or from averages of signals from two or more transmitters. See for example U.S. Pat. No. 5,594,343.
As demonstrated in U.S. Pat. Nos. 4,968,940 and 4,899,112, a very common method practiced by those skilled in the art of MWD/LWD for determining the resistivity from the measured data is to transform the dielectric constant into a variable that depends on the resistivity and then to independently convert the phase shift and attenuation measurements to two separate resistivity values. A key assumption implicitly used in this practice is that each measurement senses the resistivity within the same volume that it senses the dielectric constant. This implicit assumption is shown herein by the Applicant to be false. This currently practiced method may provide significantly incorrect resistivity values, even in virtually homogeneous earth formations; and the errors may be even more severe in inhomogeneous formations.
A MWD/LWD tool typically transmits a 2 MHz signal (although frequencies as low as 0.4 MHz are sometimes used). This frequency range is high enough to create difficulties in transforming the raw attenuation and phase measurements into accurate estimates of the resistivity and/or the dielectric constant. For example, the directly measured values are not linearly dependent on either the resistivity or the dielectric constant (this nonlinearity, known to those skilled in the art as xe2x80x9cskin-effect,xe2x80x9d also limits the penetration of the fields into the earth formation). In addition, it is useful to separate the effects of the dielectric constant and the resistivity on the attenuation and phase measurements given that both the resistivity and the dielectric constant typically vary spatially within the earth formation. If the effects of both of these variables on the measurements are not separated, the estimate of the resistivity can be corrupted by the dielectric constant, and the estimate of the dielectric constant can be corrupted by the resistivity. Essentially, the utility of separating the effects is to obtain estimates of one parameter that don""t depend on (are independent of) the other parameter. A commonly used current practice relies on assuming a correlative relationship between the resistivity and dielectric constant (i.e., to transform the dielectric constant into a variable that depends on the resistivity) and then calculating resistivity values independently from the attenuation and phase shift measurements that are consistent the correlative relationship. Differences between the resistivity values derived from corresponding phase and attenuation measurements are then ascribed to spatial variations (inhomogeneities) in the resistivity over the sensitive volume of the phase shift and attenuation measurements. See for example U.S. Pat. Nos. 4,899,112 and 4,968,940. An implicit and instrumental assumption in this method is that the attenuation measurement senses both the resistivity and dielectric constant within the same volume, and that the phase shift measurement senses both variables within the same volume (but not the same volume as the attenuation measurement). See for example U.S. Pat. Nos. 4,899,112 and, 4,968,940. These assumptions facilitate the independent determination of a resistivity value from a phase measurement and another resistivity value from an attenuation measurement. However, as is shown later, the implicit assumption mentioned above is not true; so, the results determined using such algorithms are questionable. Methods are herein disclosed to determine two resistivity values from a phase and an attenuation measurement do not use the false assumptions of the above mentioned prior art.
Another method for determining the resistivity and/or dielectric constant is to assume a model for the measurement apparatus in, for example, a homogeneous medium (no spatial variation in either the resistivity or dielectric constant) and then to determine values for the resistivity and dielectric constant that cause the model to agree with the measured phase shift and attenuation data. The resistivity and dielectric constant determined by the model are then correlated to the actual parameters of the earth formation. This method is thought to be valid only in a homogeneous medium because of the implicit assumption mentioned in the above paragraph. A recent publication by P. T. Wu, J. R. Lovell, B. Clark, S. D. Bonner, and J. R. Tabanou entitled xe2x80x9cDielectric-Independent 2-MHz Propagation Resistivitiesxe2x80x9d (SPE 56448, 1999) (hereafter referred to as xe2x80x9cWuxe2x80x9d) demonstrates that such assumptions are used by those skilled in the art. For example, Wu states: xe2x80x9cOne fundamental assumption in the computation of Rex is an uninvaded homogeneous formation. This is because the phase shift and attenuation investigate slightly different volumes.xe2x80x9d It is shown herein by Applicant that abandoning the false assumptions applied in this practice results in estimates of one parameter (i.e., the resistivity or dielectric constant) that have no net sensitivity to the other parameter. This desirable and previously unknown property of the results is very useful because earth formations are commonly inhomogeneous.
Wireline dielectric measurement tools commonly use electrical signals having frequencies in the range 20 MHz-1.1 GHz. In this range, the skin-effect is even more severe, and it is even more useful to separate the effects of the dielectric constant and resistivity. Those skilled in the art of dielectric measurements have also falsely assumed that a measurement (either attenuation or phase) senses both the resistivity and dielectric constant within the same volume. The design of the measurement equipment and interpretation of the data both reflect this. See for example U.S. Pat. Nos. 4,185,238 and 4,209,747.
Wireline induction measurements are typically not attenuation and phase, but instead the real (R) and imaginary (X) parts of the voltage across a receiver antenna which consists of several induction coils in electrical series. For the purpose of this disclosure, the R-signal for a wireline induction measurement corresponds to the phase measurement of a MWD/LWD resistivity or wireline dielectric tool, and the X-signal for a wireline induction measurement corresponds to the attenuation measurement of a MWD/LWD resistivity or wireline dielectric device. Wireline induction tools typically operate using electrical signals at frequencies from 8-200 kHz (most commonly at approximately 20 kHz). This frequency range is too low for significant dielectric sensitivity in normally encountered cases; however, the skin-effect can corrupt the wireline induction measurements. As mentioned above, the skin-effect shows up as a non-linearity in the measurement as a function of the formation conductivity, and also as a dependence of the measurement sensitivity values on the formation conductivity. Estimates of the formation conductivity from wireline induction devices are often derived from data processing algorithms which assume the tool response function is the same at all depths within the processing window. The techniques of this disclosure can be applied to wireline induction measurements for the purpose of deriving resistivity values without assuming the tool response function is the same at all depths within the processing window as is done in U.S. Pat. No. 5,157,605. In order to make such an assumption, a background conductivity, "sgr", that applies for the data within the processing window is commonly used. Practicing a disclosed embodiment reduces the dependence of the results on the accuracy of the estimates for the background parameters because the background parameters are not required to be the same at all depths within the processing window. In addition, practicing appropriate embodiments of Applicant""s techniques discussed herein reduces the need to perform steps to correct wireline induction data for the skin effect.
Techniques are provided to transform attenuation and phase measurements taken in conjunction with a drilling operation into quantities suitable for producing more accurate electrical conductivity and/or dielectric constant values. The electrical conductivity and dielectric constant values are interpreted to provide information such as the presence or absence of hydrocarbons within an earth formation penetrated by the drilling operation. The techniques can be applied to Wireline Logging, Logging-While-Drilling (LWD) and Measurement-While-Drilling (MWD) operations.
As explained above, current data processing practices in the field of MWD/LWD and wireline dielectric logging are based upon the assumption that an attenuation measurement is sensitive to the resistivity value of an earth formation in the same volume as the attenuation is sensitive to the dielectric constant. Current data processing practices are also based upon the assumption that the phase measurement is sensitive to the resistivity in the same volume of the earth formation as it is sensitive to the dielectric constant, but that this volume of the phase measurement is different from that of the attenuation measurement. These assumptions, referred to herein as the xe2x80x9cold assumptions,xe2x80x9d are shown to be false. In fact, the attenuation senses the resistivity and the dielectric constant in different volumes; and the phase shift senses the resistivity and the dielectric constant in different volumes. However, the attenuation measurement is shown to be sensitive to the resistivity in essentially the same volume as the phase measurement is sensitive to the dielectric constant. Further, the attenuation measurement is shown to be sensitive to the dielectric constant in essentially the same volume that the phase measurement is sensitive to the resistivity.
By employing these new-found relationships among the attenuation, phase, resistivity and dielectric constant, systems of simultaneous equations are provided that produce more accurate measurements of the resistivity and/or the dielectric constant within an earth formation than measurements produced using the old assumptions. In some embodiments, the equations are manipulated in a manner that provides a resistivity component that is relatively insensitive to the dielectric constant and provides a dielectric constant component that is relatively insensitive to the resistivity value. Thus, more accurate and/or robust calculations of both the resistivity and the dielectric constant are produced.
The disclosed techniques are also applied to more complicated scenarios wherein multiple transmitters (possibly driven at multiple frequencies), multiple receivers, data acquired at multiple depths, or combinations of the above are considered simultaneously. Solving a prescribed system of equations using this disclosed embodiment results in estimates of an average conductivity value and an average dielectric constant value within a volume of the earth formation corresponding to integrated averages of each parameter over said volume. In general, the resistivity and dielectric constant are expanded using basis functions to characterize the spatial dependence of these variables. A system of equations which can be solved for the coefficients of this expansion is given. Once the coefficients are determined, the spatial dependence of both the resistivity and dielectric constant are known.
One disclosed embodiment employs a transformation to convert the dielectric constant into a variable that depends on the resistivity thereby eliminating the dielectric constant as a variable. Two resistivity estimates from a phase shift and an attenuation measurement are then calculated. These estimates are not determined independently as is done in the prior art because the equations solved to obtain the estimates are coupled. The manner in which these equations are coupled is consistent with the actual sensitivities of each measurement (i.e., the phase shift and attenuation) with respect to changes in each variable (i.e., the resistivity and dielectric constant). Unlike previous MWD/LWD processing techniques, some disclosed embodiments account for dielectric effects, provide for inhomogeneities, and treat each signal as a complex-valued function of the conductivity and dielectric constant, assuring that estimates of each variable are not corrupted by effects of the other. Treating both the measurements (attenuation and phase) and the variables (conductivity and dielectric constant) mathematically as complex-valued functions is a useful feature of the disclosed embodiments. Good results from the disclosed embodiments are produced by using the new found relationship regarding the volume of investigation of each measurement with respect to the conductivity and the dielectric constant. In contrast, the old assumptions imply that these results are impracticable. This is readily evident from discussions in U.S. Pat. Nos. 4,185,238; 4,209,747; 4,899,112; and 4,968,940.